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The Compatibility of Differential Equations and Causal Models Reconsidered

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Abstract

Weber argues that causal modelers face a dilemma when they attempt to model systems in which the underlying mechanism operates according to some set of differential equations. The first horn is that causal models of these systems leave out certain causal effects. The second horn is that causal models of these systems leave out time-dependent derivatives, and doing so distorts reality. Either way causal models of these systems leave something important out. I argue that Weber’s reasons for thinking causal modeling is limited in this domain are lacking.

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Notes

  1. But see, for example, Maier (2014) for one of the most recent and important attempts to handle non-iid instances.

  2. Weber also considers what he calls “push and let go” interventions where you hold a variable to a value for a brief amount of time and then let it go. Here he says these interventions would also not allow us to predict causal effects. I do not consider these in this paper.

  3. The do operator is primarily used in the do-calculus which is sound and complete for identifying identifiable causal effects non-parametricaly. If there were a restiction on the instantiated form of f or the parameter values in f given an application of the do operator, then the do-calculus would not be complete. Hence, this must be the way to interpret Pearl’s do operator. I thank an anonymous referee for suggestions pertaining to this point.

  4. I thank an anonymous referee for helping me clarify my point here.

  5. We could not assign it to \(F_t\), because then the equation for \(F_t\), i.e., \(F_t = Mg\), would have to be assigned to M, but then we would be left with the equation \(M=m\), and hence, M would have two equations. But this violates our one-to-one correspondence assumption. There are of course systems where this assumption is violated, and hence such systems have no single equilibration. But I simply do not deal with these here.

  6. I thank an anonymous referee and Weber (personal communication) for helping me clarify this point.

References

  • Aalen, O., Røysland, K., Gran, J., & Ledergerber, B. (2012). Causality, mediation, and time: A dynamic viewpoint. Journal of the Royal Statistical Society: Series A (Statistics in Society), 175, 831–861.

    Article  Google Scholar 

  • Dash, D. (2003). Caveats for causal reasoning with equilibrium models. University of Pittsburgh Ph.D. Dissertation.

  • Glymour, B. (2011). Modeling environments: Interactive causation and adaptations to environmental conditions. Philosophy of Science, 78, 448–471.

    Article  Google Scholar 

  • Glymour, C. (2008). When is a brain like the planet? Philosophy of Science, 74, 330–347.

    Article  Google Scholar 

  • Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). In Proceedings of the Royal Society of London. Part B: Biological Sciences (Vol. 261, pp. 319–324).

  • Hyttinen, A., Plis, S., Järvisalo, M., Eberhardt, F., & Danks, D. (2016). Causal discovery from subsampled time series data by constraint optimization. Journal of Machine Learning Research Workshop and Conference Proceedings, 52, 216–227.

    Google Scholar 

  • Iwasaki, Y., & Simon, H. A. (1994). Causality and model abstraction. Artificial Intelligence, 67, 143–194.

    Article  Google Scholar 

  • Jantzen, B. (2015). Projection, symmetry, and natural kinds. Synthese, 192, 3617–3646.

    Article  Google Scholar 

  • Kuorikoski, J. (2012). Mechanisms, modularity, and constitutive explanation. Erkenntnis, 78, 1–20.

    Google Scholar 

  • Maier, M. (2014). Causal discovery for relational domains: Representation, reasoning, and learning. Ph.D. Dissertation. University of Massachusetts Amherst.

  • Mooij, J. M., Janzing, D., & Schölkopf, B. (2013). From ordinary differential equations to structural causal models: The deterministic case. In Proceedings of the 29th Annual Conference on Uncertainty in Artificial Intelligence (pp. 440–448).

  • Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge: Cambridge University Press.

    Google Scholar 

  • Plis, S., Danks, D., & Yang, J. (2015). Mesochronal structure learning. In Uncertainty in artificial intelligence (Vol. 31, pp. 702–711).

  • Sokol, A. (2013). On martingales, causality, identifiability, and model selection. University of Copenhagen Ph.D. Dissertation.

  • Spirtes, P., Glymour, C., & Scheines, R. (2000). Causation, prediction, and search. Cambridge, MA: MIT Press.

    Google Scholar 

  • Voortman, M., Dash, D., & Druzdzel, M. J. (2010). Learning why things change: The difference-based causality learner. In Proceedings of the 26th Annual Conference on Uncertainty in Artificial Intelligence (pp. 641–650).

  • Weber, M. (2016). On the incompatibility of dynamical biological mechanisms and causal graphs. Philosophy of Science, 83, 959–971.

    Article  Google Scholar 

  • Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.

    Google Scholar 

Download references

Acknowledgements

I wish to thank James DiFrisco, Bruce Glymour, Valerie Racine, Marcel Weber, and two anonymous referees for various helpful suggestions on previous drafts of this paper.

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Correspondence to Wes Anderson.

Appendix: Classical Causal Inference

Appendix: Classical Causal Inference

The causal Markov axiom implies a set of independence claims about a set of causally sufficient variables \(\mathbf{V}\) on a directed acyclic causal graph \(\mathcal {G}\). To state this, say that if \(X=Y\) or there is a directed path from X to Y, then X is a graphical ancestor or cause of Y and Y is a graphical descendant or effect of X. Then

Definition 1

(causal Markov) The joint probability distribution over a causally sufficient set of variables \(\mathbf{V}\) is causally Markov to directed acyclic causal graph \(\mathcal {G}\) over \(\mathbf{V}\) just in case each variable \(V \in \mathbf{V}\) is independent of its graphical non-descendants conditional on its graphical parents.

Conversely, the faithfulness axiom can be stated in the following way

Definition 2

(faithfulness) The joint probability distribution over \(\mathbf{V}\) is faithful to \(\mathcal {G}\) over \(\mathbf{V}\) just in case there are no conditional independencies in the joint probability distribution that are not entailed by the causal Markov axiom on \(\mathcal {G}\).

The causal Markov and faithfulness axioms allow us to, respectively, go from d-separation facts to claims of conditional independence and d-connection facts to claims of conditional dependence.

The graphical criteria of d-separation, in turn, allows us to go from \(\mathcal {G}\) to d-separation and d-connection claims about variables on \(\mathcal {G}\). All that is required to get us there is the graphical idea of a collider. A variable Z is a collider on a path between X and Y just in case the edges incident Z on the path are into Z. Now

Definition 3

(d-separation) Let \(\mathbf{X}\), \(\mathbf{Y}\), and \(\mathbf{Z}\) be three disjoint sets of variables of \(\mathbf{V}\) on \(\mathcal {G}\). Sets of variables \(\mathbf{X}\) and \(\mathbf{Y}\) are d-separated by \(\mathbf{Z}\) just in case, for all \(X \in \mathbf{X}\) and all \(Y \in \mathbf{Y}\), there is no path between X and Y on \(\mathcal {G}\) in which

  1. 1.

    every collider has a descendant in \(\mathbf{Z}\) and

  2. 2.

    no non-collider is in \(\mathbf{Z}\).

In other words two sets of variables \(\mathbf{X}\) and \(\mathbf{Y}\) are d-separated by a third set \(\mathbf{Z}\) just in case there is no d-connecting path between any \(X \in \mathbf{X}\) and any \(Y \in \mathbf{Y}\) on \(\mathcal {G}\).

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Anderson, W. The Compatibility of Differential Equations and Causal Models Reconsidered. Erkenn 85, 317–332 (2020). https://doi.org/10.1007/s10670-018-0029-1

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